Characteristic classes and transversality

Let ξ be a smooth vector bundle over a differentiable manifold M. Let h : ε^{n - i + 1} → ξ be a generic bundle morphism from the trivial bundle of rank n - i + 1 to ξ. We give a geometric construction of the Stiefel-Whitney classes when ξ is a real vector bundle, and of the Chern classes when ξ is a complex vector bundle. Using h we define a differentiable closed manifold over(Z, ̃) (h) and a map φ{symbol} : over(Z, ̃) (h) → M whose image is the singular set of h. The ith characteristic class of ξ is the Poincaré dual of the image, under the homomorphism induced in homology by φ{symbol}, of the fundamental class of the manifold over(Z, ̃) (h). We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.